General Relativity : An Introduction for PhysicistsM._P...General Relativity: An Introduction for...

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General Relativity: An Introduction for Physicists provides a clear mathematicalintroduction to Einstein’s theory of general relativity. A wide range of applicationsof the theory are included, with a concentration on its physical consequences.

After reviewing the basic concepts, the authors present a clear and intuitivediscussion of the mathematical background, including the necessary tools of tensorcalculus and differential geometry. These tools are used to develop the topicof special relativity and to discuss electromagnetism in Minkowski spacetime.Gravitation as spacetime curvature is then introduced and the field equations ofgeneral relativity are derived. A wide range of applications to physical situationsfollows, and the conclusion gives a brief discussion of classical field theory andthe derivation of general relativity from a variational principle.

Written for advanced undergraduate and graduate students, this approachabletextbook contains over 300 exercises to illuminate and extend the discussion inthe text.

Michael Hobson specialised in theoretical physics as an undergraduate at theUniversity of Cambridge and remained at the Cavendish Laboratory to completea Ph.D. in the physics of star-formation and radiative transfer. As a ResearchFellow at Trinity Hall, Cambridge, and later as an Advanced Fellow of the ParticlePhysics and Astronomy Research Council, he developed an interest in cosmology,in particular in the study of fluctuations in the cosmic microwave background(CMB) radiation. He is currently a Reader in Astrophysics and Cosmology at theCavendish Laboratory, where he is the principal investigator for the Very SmallArray CMB interferometer. He is also joint project scientist for the ArcminuteMicrokelvin Imager project and an associate of the European Space AgencyPlanck Surveyor CMB satellite mission. In addition to observational and theo-retical cosmology, his research interests also include Bayesian analysis methodsand theoretical optics and he has published over 100 research papers in a widerange of areas. He is a Staff Fellow and Director of Studies in Natural Sciencesat Trinity Hall and enjoys an active role in the teaching of undergraduate physicsand mathematics. He is a co-author with Ken Riley and Stephen Bence of thewell-known undergraduate textbookMathematical Methods for Physics and Engi-neering (Cambridge, 1998; second edition, 2002; third edition to be published in2006) and with Ken Riley of the Student’s Solutions Manual accompanying thethird edition.

George Efstathiou is Professor of Astrophysics and Director of the Instituteof Astronomy at the University of Cambridge. After studying physics as anundergraduate at Keble College, Oxford, he gained his Ph.D. in astronomy fromDurham University. Following some post-doctoral research at the University of

California at Berkeley he returned to work in the UK at the Institute of Astronomy,Cambridge, where he was appointed Assistant Director of Research in 1987.He returned to the Department of Physics at Oxford as Savilian Professor ofAstronomy and Head of Astrophysics, before taking on his current posts at theInstitute of Astronomy in 1997 and 2004 respectively. He is a Fellow of the RoyalSociety and the recipient of several awards, including the Maxwell Medal andPrize of the Institute of Physics in 1990 and the Heineman Prize for Astronomyof the American Astronomical Society in 2005.

Anthony Lasenby is Professor of Astrophysics and Cosmology at the Universityof Cambridge and is currently Head of the Astrophysics Group and the MullardRadio Astronomy Observatory in the Cavendish Laboratory, as well as being aDeputy Head of the Laboratory. He began his astronomical career with a Ph.D.at Jodrell Bank, specializing in the cosmic microwave background, which hasremained a major subject of his research. After a brief period at the National RadioAstronomy Observatory in America, he moved from Manchester to Cambridge in1984 and has been at the Cavendish since then. He is the author or co-author ofover 200 papers spanning a wide range of fields and is the co-author of GeometricAlgebra for Physicists (Cambridge, 2003) with Chris Doran.

General RelativityAn Introduction for Physicists

M . P . HOB SON , G . P . E F S TATH IOUand A . N . LA S ENBY

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Contents

Preface page xv

1 The spacetime of special relativity 11.1 Inertial frames and the principle of relativity 11.2 Newtonian geometry of space and time 31.3 The spacetime geometry of special relativity 31.4 Lorentz transformations as four-dimensional ‘rotations’ 51.5 The interval and the lightcone 61.6 Spacetime diagrams 81.7 Length contraction and time dilation 101.8 Invariant hyperbolae 111.9 The Minkowski spacetime line element 121.10 Particle worldlines and proper time 141.11 The Doppler effect 161.12 Addition of velocities in special relativity 181.13 Acceleration in special relativity 191.14 Event horizons in special relativity 21Appendix 1A: Einstein’s route to special relativity 22Exercises 24

2 Manifolds and coordinates 262.1 The concept of a manifold 262.2 Coordinates 272.3 Curves and surfaces 272.4 Coordinate transformations 282.5 Summation convention 302.6 Geometry of manifolds 312.7 Riemannian geometry 322.8 Intrinsic and extrinsic geometry 33

vii

viii Contents

2.9 Examples of non-Euclidean geometry 362.10 Lengths, areas and volumes 382.11 Local Cartesian coordinates 422.12 Tangent spaces to manifolds 442.13 Pseudo-Riemannian manifolds 452.14 Integration over general submanifolds 472.15 Topology of manifolds 49Exercises 50

3 Vector calculus on manifolds 533.1 Scalar fields on manifolds 533.2 Vector fields on manifolds 543.3 Tangent vector to a curve 553.4 Basis vectors 563.5 Raising and lowering vector indices 593.6 Basis vectors and coordinate transformations 603.7 Coordinate-independent properties of vectors 613.8 Derivatives of basis vectors and the affine connection 623.9 Transformation properties of the affine connection 643.10 Relationship of the connection and the metric 653.11 Local geodesic and Cartesian coordinates 673.12 Covariant derivative of a vector 683.13 Vector operators in component form 703.14 Intrinsic derivative of a vector along a curve 713.15 Parallel transport 733.16 Null curves, non-null curves and affine parameters 753.17 Geodesics 763.18 Stationary property of non-null geodesics 773.19 Lagrangian procedure for geodesics 783.20 Alternative form of the geodesic equations 81Appendix 3A: Vectors as directional derivatives 81Appendix 3B: Polar coordinates in a plane 82Appendix 3C: Calculus of variations 87Exercises 88

4 Tensor calculus on manifolds 924.1 Tensor fields on manifolds 924.2 Components of tensors 934.3 Symmetries of tensors 944.4 The metric tensor 964.5 Raising and lowering tensor indices 97

Contents ix

4.6 Mapping tensors into tensors 974.7 Elementary operations with tensors 984.8 Tensors as geometrical objects 1004.9 Tensors and coordinate transformations 1014.10 Tensor equations 1024.11 The quotient theorem 1034.12 Covariant derivative of a tensor 1044.13 Intrinsic derivative of a tensor along a curve 107Exercises 108

5 Special relativity revisited 1115.1 Minkowski spacetime in Cartesian coordinates 1115.2 Lorentz transformations 1125.3 Cartesian basis vectors 1135.4 Four-vectors and the lightcone 1155.5 Four-vectors and Lorentz transformations 1165.6 Four-velocity 1165.7 Four-momentum of a massive particle 1185.8 Four-momentum of a photon 1195.9 The Doppler effect and relativistic aberration 1205.10 Relativistic mechanics 1225.11 Free particles 1235.12 Relativistic collisions and Compton scattering 1235.13 Accelerating observers 1255.14 Minkowski spacetime in arbitrary coordinates 128Exercises 131

6 Electromagnetism 1356.1 The electromagnetic force on a moving charge 1356.2 The 4-current density 1366.3 The electromagnetic field equations 1386.4 Electromagnetism in the Lorenz gauge 1396.5 Electric and magnetic fields in inertial frames 1416.6 Electromagnetism in arbitrary coordinates 1426.7 Equation of motion for a charged particle 144Exercises 145

7 The equivalence principle and spacetime curvature 1477.1 Newtonian gravity 1477.2 The equivalence principle 1487.3 Gravity as spacetime curvature 1497.4 Local inertial coordinates 151

x Contents

7.5 Observers in a curved spacetime 1527.6 Weak gravitational fields and the Newtonian limit 1537.7 Electromagnetism in a curved spacetime 1557.8 Intrinsic curvature of a manifold 1577.9 The curvature tensor 1587.10 Properties of the curvature tensor 1597.11 The Ricci tensor and curvature scalar 1617.12 Curvature and parallel transport 1637.13 Curvature and geodesic deviation 1657.14 Tidal forces in a curved spacetime 167Appendix 7A: The surface of a sphere 170Exercises 172

8 The gravitational field equations 1768.1 The energy–momentum tensor 1768.2 The energy–momentum tensor of a perfect fluid 1788.3 Conservation of energy and momentum for a perfect fluid 1798.4 The Einstein equations 1818.5 The Einstein equations in empty space 1838.6 The weak-field limit of the Einstein equations 1848.7 The cosmological-constant term 1858.8 Geodesic motion from the Einstein equations 1888.9 Concluding remarks 190Appendix 8A: Alternative relativistic theories of gravity 191Appendix 8B: Sign conventions 193Exercises 193

9 The Schwarzschild geometry 1969.1 The general static isotropic metric 1969.2 Solution of the empty-space field equations 1989.3 Birkhoff’s theorem 2029.4 Gravitational redshift for a fixed emitter and receiver 2029.5 Geodesics in the Schwarzschild geometry 2059.6 Trajectories of massive particles 2079.7 Radial motion of massive particles 2099.8 Circular motion of massive particles 2129.9 Stability of massive particle orbits 2139.10 Trajectories of photons 2179.11 Radial motion of photons 2189.12 Circular motion of photons 2199.13 Stability of photon orbits 220

Contents xi

Appendix 9A: General approach to gravitational redshifts 221Exercises 224

10 Experimental tests of general relativity 23010.1 Precession of planetary orbits 23010.2 The bending of light 23310.3 Radar echoes 23610.4 Accretion discs around compact objects 24010.5 The geodesic precession of gyroscopes 244Exercises 246

11 Schwarzschild black holes 24811.1 The characterisation of coordinates 24811.2 Singularities in the Schwarzschild metric 24911.3 Radial photon worldlines in Schwarzschild coordinates 25111.4 Radial particle worldlines in Schwarzschild coordinates 25211.5 Eddington–Finkelstein coordinates 25411.6 Gravitational collapse and black-hole formation 25911.7 Spherically symmetric collapse of dust 26011.8 Tidal forces near a black hole 26411.9 Kruskal coordinates 26611.10 Wormholes and the Einstein–Rosen bridge 27111.11 The Hawking effect 274Appendix 11A: Compact binary systems 277Appendix 11B: Supermassive black holes 279Appendix 11C: Conformal flatness of two-dimensional Riemannian

manifolds 282Exercises 283

12 Further spherically symmetric geometries 28812.1 The form of the metric for a stellar interior 28812.2 The relativistic equations of stellar structure 29212.3 The Schwarzschild constant-density interior solution 29412.4 Buchdahl’s theorem 29612.5 The metric outside a spherically symmetric

charged mass 29612.6 The Reissner–Nordström geometry: charged

black holes 30012.7 Radial photon trajectories in the RN geometry 30212.8 Radial massive particle trajectories

in the RN geometry 304Exercises 305

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13 The Kerr geometry 31013.1 The general stationary axisymmetric metric 31013.2 The dragging of inertial frames 31213.3 Stationary limit surfaces 31413.4 Event horizons 31513.5 The Kerr metric 31713.6 Limits of the Kerr metric 31913.7 The Kerr–Schild form of the metric 32113.8 The structure of a Kerr black hole 32213.9 The Penrose process 32713.10 Geodesics in the equatorial plane 33013.11 Equatorial trajectories of massive particles 33213.12 Equatorial motion of massive particles with

zero angular momentum 33313.13 Equatorial circular motion of massive particles 33513.14 Stability of equatorial massive particle circular orbits 33713.15 Equatorial trajectories of photons 33813.16 Equatorial principal photon geodesics 33913.17 Equatorial circular motion of photons 34113.18 Stability of equatorial photon orbits 34213.19 Eddington–Finkelstein coordinates 34413.20 The slow-rotation limit and gyroscope precession 347Exercises 350

14 The Friedmann–Robertson–Walker geometry 35514.1 The cosmological principle 35514.2 Slicing and threading spacetime 35614.3 Synchronous coordinates 35714.4 Homogeneity and isotropy of the universe 35814.5 The maximally symmetric 3-space 35914.6 The Friedmann–Robertson–Walker metric 36214.7 Geometric properties of the FRW metric 36214.8 Geodesics in the FRW metric 36514.9 The cosmological redshift 36714.10 The Hubble and deceleration parameters 36814.11 Distances in the FRW geometry 37114.12 Volumes and number densities in the FRW geometry 37414.13 The cosmological field equations 37614.14 Equation of motion for the cosmological fluid 37914.15 Multiple-component cosmological fluid 381Exercises 381

Contents xiii

15 Cosmological models 38615.1 Components of the cosmological fluid 38615.2 Cosmological parameters 39015.3 The cosmological field equations 39215.4 General dynamical behaviour of the universe 39315.5 Evolution of the scale factor 39715.6 Analytical cosmological models 40015.7 Look-back time and the age of the universe 40815.8 The distance–redshift relation 41115.9 The volume–redshift relation 41315.10 Evolution of the density parameters 41515.11 Evolution of the spatial curvature 41715.12 The particle horizon, event horizon and Hubble distance 418Exercises 421

16 Inflationary cosmology 42816.1 Definition of inflation 42816.2 Scalar fields and phase transitions in the very early universe 43016.3 A scalar field as a cosmological fluid 43116.4 An inflationary epoch 43316.5 The slow-roll approximation 43416.6 Ending inflation 43516.7 The amount of inflation 43516.8 Starting inflation 43716.9 ‘New’ inflation 43816.10 Chaotic inflation 44016.11 Stochastic inflation 44116.12 Perturbations from inflation 44216.13 Classical evolution of scalar-field perturbations 44216.14 Gauge invariance and curvature perturbations 44616.15 Classical evolution of curvature perturbations 44916.16 Initial conditions and normalisation of curvature perturbations 45216.17 Power spectrum of curvature perturbations 45616.18 Power spectrum of matter-density perturbations 45816.19 Comparison of theory and observation 459Exercises 462

17 Linearised general relativity 46717.1 The weak-field metric 46717.2 The linearised gravitational field equations 47017.3 Linearised gravity in the Lorenz gauge 472

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17.4 General properties of the linearised field equations 47317.5 Solution of the linearised field equations in vacuo 47417.6 General solution of the linearised field equations 47517.7 Multipole expansion of the general solution 48017.8 The compact-source approximation 48117.9 Stationary sources 48317.10 Static sources and the Newtonian limit 48517.11 The energy–momentum of the gravitational field 486Appendix17A:TheEinstein–Maxwell formulationof linearisedgravity 490Exercises 493

18 Gravitational waves 49818.1 Plane gravitational waves and polarisation states 49818.2 Analogy between gravitational and electromagnetic waves 50118.3 Transforming to the transverse-traceless gauge 50218.4 The effect of a gravitational wave on free particles 50418.5 The generation of gravitational waves 50718.6 Energy flow in gravitational waves 51118.7 Energy loss due to gravitational-wave emission 51318.8 Spin-up of binary systems: the binary pulsar PSR B1913+16 51618.9 The detection of gravitational waves 517Exercises 520

19 A variational approach to general relativity 52419.1 Hamilton’s principle in Newtonian mechanics 52419.2 Classical field theory and the action 52719.3 Euler–Lagrange equations 52919.4 Alternative form of the Euler–Lagrange equations 53119.5 Equivalent actions 53319.6 Field theory of a real scalar field 53419.7 Electromagnetism from a variational principle 53619.8 The Einstein–Hilbert action and general relativity in vacuo 53919.9 An equivalent action for general relativity in vacuo 54219.10 The Palatini approach for general relativity in vacuo 54319.11 General relativity in the presence of matter 54519.12 The dynamical energy–momentum tensor 546Exercises 549

Bibliography 555

Index 556

Preface

General relativity is one of the cornerstones of classical physics, providing asynthesis of special relativity and gravitation, and is central to our understandingof many areas of astrophysics and cosmology. This book is intended to give anintroduction to this important subject, suitable for a one-term course for advancedundergraduate or beginning graduate students in physics or in related disciplinessuch as astrophysics and applied mathematics. Some of the later chapters shouldalso provide a useful reference for professionals in the fields of astrophysics andcosmology.

It is assumed that the reader has already been exposed to special relativity andNewtonian gravitation at a level typical of early-stage university physics courses.Nevertheless, a summary of special relativity from first principles is given inChapter 1, and a brief discussion of Newtonian gravity is presented in Chapter 7.No previous experience of 4-vector methods is assumed. Some background inelectromagnetism will prove useful, as will some experience of standard vectorcalculus methods in three-dimensional Euclidean space. The overall level of math-ematical expertise assumed is that of a typical university mathematical methodscourse.

The book begins with a review of the basic concepts underlying special rela-tivity in Chapter 1. The subject is introduced in a way that encourages from theoutset a geometrical and transparently four-dimensional viewpoint, which lays theconceptual foundations for discussion of the more complicated spacetime geome-tries encountered later in general relativity. In Chapters 2–4 we then present amini-course in basic differential geometry, beginning with the introduction ofmanifolds, coordinates and non-Euclidean geometry in Chapter 2. The topic ofvector calculus on manifolds is developed in Chapter 3, and these ideas areextended to general tensors in Chapter 4. These necessary mathematical prelimi-naries are presented in such a way as to make them accessible to physics studentswith a background in standard vector calculus. A reasonable level of mathematical

xv

xvi Preface

rigour has been maintained throughout, albeit accompanied by the occasionalappeal to geometric intuition. The mathematical tools thus developed are thenillustrated in Chapter 5 by re-examining the familiar topic of special relativity in amore formal manner, through the use of tensor calculus in Minkowski spacetime.These methods are further illustrated in Chapter 6, in which electromagnetism isdescribed as a field theory in Minkowski spacetime, serving in some respects as a‘prototype’ for the later discussion of gravitation. In Chapter 7, the incompatibilityof special relativity and Newtonian gravitation is presented and the equivalenceprinciple is introduced. This leads naturally to a discussion of spacetime curvatureand the associated mathematics. The field equations of general relativity are thenderived in Chapter 8, and a discussion of their general properties is presented.

The physical consequences of general relativity in a wide variety of astrophys-ical and cosmological applications are discussed in Chapters 9–18. In particular,the Schwarzschild geometry is derived in Chapter 9 and used to discuss the physicsoutside a massive spherical body. Classic experimental tests of general relativitybased on the exterior Schwarzschild geometry are presented in Chapter 10. Theinterior Schwarzschild geometry and non-rotating black holes are discussed inChapter 11, together with a brief mention of Kruskal coordinates and wormholes.In Chapter 12 we introduce two non-vacuum spherically symmetric geometrieswith a discussion of relativistic stars and charged black holes. Rotating objects arediscussed in Chapter 13, including an extensive discussion of the Kerr solution. InChapters 14–16 we describe the application of general relativity to cosmology andpresent a discussion of the Friedmann–Robertson–Walker geometry, cosmologi-cal models and the theory of inflation, including the generation of perturbationsin the early universe. In Chapter 17 we describe linearised gravitation and weakgravitational fields, in particular drawing analogies with the theory of electromag-netism. The equations of linearised gravitation are then applied to the generation,propagation and detection of weak gravitational waves in Chapter 18. The bookconcludes in Chapter 19 with a brief discussion of classical field theory and thederivation of the field equations of electromagnetism and general relativity fromvariational principles.

Each chapter concludes with a number of exercises that are intended to illumi-nate and extend the discussion in the main text. It is strongly recommended thatthe reader attempt as many of these exercises as time permits, as they should giveample opportunity to test his or her understanding. Occasionally chapters haveappendices containing material that is not central to the development presented inthe main text, but may nevertheless be of interest to the reader. Some appendicesprovide historical context, some discuss current astronomical observations andsome give detailed mathematical derivations that might otherwise interrupt theflow of the main text.

Preface xvii

With regard to the presentation of the mathematics, it has to be acceptedthat equations containing partial and covariant derivatives could be written morecompactly by using the comma and semi-colon notation, e.g. va�b for the partialderivative of a vector and va�b for its covariant derivative. This would certainlysave typographical space, but many students find the labour of mentally unpackingsuch equations is sufficiently great that it is not possible to think of an equation’sphysical interpretation at the same time. Consequently, we have decided to writeout such expressions in their more obvious but longer form, using �bv

a for partialderivatives and �bv

a for covariant derivatives.It is worth mentioning that this book is based, in large part, on lecture notes

prepared separately by MPH and GPE for two different relativity courses in theNatural Science Tripos at the University of Cambridge. These courses were firstpresented in this form in the academic year 1999–2000 and are still ongoing. Thecourse presented by MPH consisted of 16 lectures to fourth-year undergraduatesin Part III Physics and Theoretical Physics and covered most of the materialin Chapters 1–11 and 13–14, albeit somewhat rapidly on occasion. The coursegiven by GPE consisted of 24 lectures to third-year undergraduates in Part IIAstrophysics and covered parts of Chapters 1, 5–11, 14 and 18, with an emphasison the less mathematical material. The process of combining the two sets oflecture notes into a homogeneous treatment of relativistic gravitation was aidedsomewhat by the fortuitous choice of a consistent sign convention in the twocourses, and numerous sections have been rewritten in the hope that the readerwill not encounter any jarring changes in presentational style. For many of thetopics covered in the two courses mentioned above, the opportunity has beentaken to include in this book a considerable amount of additional material beyondthat presented in the lectures, especially in the discussion of black holes. Someof this material draws on lecture notes written by ANL for other courses in PartII and Part III Physics and Theoretical Physics. Some topics that were entirelyabsent from any of the above lecture courses have also been included in the book,such as relativistic stars, cosmology, inflation, linearised gravity and variationalprinciples. While every care has been taken to describe these topics in a clear andilluminating fashion, the reader should bear in mind that these chapters have notbeen ‘road-tested’ to the same extent as the rest of the book.

It is with pleasure that we record here our gratitude to those authors fromwhose books we ourselves learnt general relativity and who have certainlyinfluenced our own presentation of the subject. In particular, we acknowledge(in their current latest editions) S. Weinberg, Gravitation and Cosmology,Wiley, 1972; R. M. Wald, General Relativity, University of Chicago Press,1984; B. Schutz, A First Course in General Relativity, Cambridge Univer-sity Press, 1985; W. Rindler, Relativity: Special, General and Cosmological,

xviii Preface

Oxford University Press, 2001; and J. Foster & J. D. Nightingale, A Short Coursein General Relativity, Springer-Verlag, 1995.

During the writing of this book we have received much help and encourage-ment from many of our colleagues at the University of Cambridge, especiallymembers of the Cavendish Astrophysics Group and the Institute of Astronomy.In particular, we thank Chris Doran, Anthony Challinor, Steve Gull and PaulAlexander for numerous useful discussions on all aspects of relativity theory, andDave Green for a great deal of advice concerning typesetting in LaTeX. We arealso especially grateful to Richard Sword for creating many of the diagrams andfigures used in the book and to Michael Bridges for producing the plots of recentmeasurements of the cosmic microwave background and matter power spectra.We also extend our thanks to the Cavendish and Institute of Astronomy teach-ing staff, whose examination questions have provided the basis for some of theexercises included. Finally, we thank several years of undergraduate students fortheir careful reading of sections of the manuscript, for pointing out misprints andfor numerous useful comments. Of course, any errors and ambiguities remainingare entirely the responsibility of the authors, and we would be most grateful tohave them brought to our attention. At Cambridge University Press, we are verygrateful to our editor Vince Higgs for his help and patience and to our copy-editorSusan Parkinson for many useful suggestions that have undoubtedly improved thestyle of the book.

Finally, on a personal note, MPH thanks his wife, Becky, for patiently enduringmany evenings and weekends spent listening to the sound of fingers tapping ona keyboard, and for her unending encouragement. He also thanks his mother,Pat, for her tireless support at every turn. MPH dedicates his contribution to thisbook to the memory of his father, Ron, and to his daughter, Tabitha, whose earlyarrival succeeded in delaying completion of the book by at least three months, butequally made him realise how little that mattered. GPE thanks his wife, Yvonne,for her support. ANL thanks all the students who have sat through his variouslectures on gravitation and cosmology and provided useful feedback. He wouldalso like to thank his family, and particularly his parents, for the encouragementand support they have offered at all times.

1

The spacetime of special relativity

We begin our discussion of the relativistic theory of gravity by reviewing somebasic notions underlying the Newtonian and special-relativistic viewpoints ofspace and time. In order to specify an event uniquely, we must assign it threespatial coordinates and one time coordinate, defined with respect to some frameof reference. For the moment, let us define such a system S by using a set of threemutually orthogonal Cartesian axes, which gives us spatial coordinates x, y andz, and an associated system of synchronised clocks at rest in the system, whichgives us a time coordinate t. The four coordinates �t� x� y� z� thus label events inspace and time.

1.1 Inertial frames and the principle of relativity

Clearly, one is free to label events not only with respect to a frame S but alsowith respect to any other frame S′, which may be oriented and/or moving withrespect to S in an arbitrary manner. Nevertheless, there exists a class of preferredreference systems called inertial frames, defined as those in which Newton’s firstlaw holds, so that a free particle is at rest or moves with constant velocity, i.e. ina straight line with fixed speed. In Cartesian coordinates this means that

d2x

dt2= d

2y

dt2= d

2z

dt2= 0�

It follows that, in the absence of gravity, if S and S′ are two inertial frames thenS′ can differ from S only by (i) a translation, and/or (ii) a rotation and/or (iii) amotion of one frame with respect to the other at a constant velocity (for otherwiseNewton’s first law would no longer be true). The concept of inertial frames isfundamental to the principle of relativity, which states that the laws of physicstake the same form in every inertial frame. No exception has ever been found to

1

2 The spacetime of special relativity

this general principle, and it applies equally well in both Newtonian theory andspecial relativity.

The Newtonian and special-relativistic descriptions differ in how the coor-dinates of an event P in two inertial frames are related. Let us consider twoCartesian inertial frames S and S′ in standard configuration, where S′ is movingalong the x-axis of S at a constant speed v and the axes of S and S′ coincide att = t′ = 0 (see Figure 1.1). It is clear that the (primed) coordinates of an eventP with respect to S′ are related to the (unprimed) coordinates in S via a lineartransformation1 of the form

t′ = At+Bx�x′ = Dt+Ex�y′ = y�z′ = z�

Moreover, since we require that x′ = 0 corresponds to x = vt and that x = 0corresponds to x′ = −vt′, we find immediately that D =−Ev and D =−Av, sothat A= E. Thus we must have

t′ = At+Bx�x′ = A�x−vt��y′ = y�z′ = z�

(1.1)

x x'

y'y

S S'

z'z

v

Figure 1.1 Two inertial frames S and S′ in standard configuration (the originsof S and S′ coincide at t = t′ = 0).

1 We will prove this in Chapter 5.

1.3 The spacetime geometry of special relativity 3

1.2 Newtonian geometry of space and time

Newtonian theory rests on the assumption that there exists an absolute time, whichis the same for every observer, so that t′ = t. Under this assumption A = 1 andB = 0, and we obtain the Galilean transformation relating the coordinates of anevent P in the two Cartesian inertial frames S and S′:

t′ = t�x′ = x−vt�y′ = y�z′ = z�

(1.2)

By symmetry, the expressions for the unprimed coordinates in terms of the primedones have the same form but with v replaced by −v.

The first equation in (1.2) is clearly valid for any two inertial frames S andS′ and shows that the time coordinate of an event P is the same in all inertialframes. The second equation leads to the ‘common sense’ notion of the additionof velocities. If a particle is moving in the x-direction at a speed u in S then itsspeed in S′ is given by

u′x =dx′

dt′= dx

′

dt= dx

dt−v= ux−v�

Differentiating again shows that the acceleration of a particle is the same in bothS and S′, i.e. du′x/dt′ = dux/dt.

If we consider two events A and B that have coordinates �tA� xA� yA� zA�and �tB� xB� yB� zB� respectively, it is straightforward to show that both the timedifference �t = tB− tA and the quantity

�r2 = �x2+�y2+�z2

are separately invariant under any Galilean transformation. This leads us toconsider space and time as separate entities. Moreover, the invariance of �r2

suggests that it is a geometric property of space itself. Of course, we recognise�r2 as the square of the distance between the events in a three-dimensionalEuclidean space. This defines the geometry of space and time in the Newtonianpicture.

1.3 The spacetime geometry of special relativity

In special relativity, Einstein abandoned the postulate of an absolute time andreplaced it by the postulate that the speed of light c is the same in all inertial


frames.2 By applying this new postulate, together with the principle of relativity,we may obtain the Lorentz transformations connecting the coordinates of an eventP in two different Cartesian inertial frames S and S′.

Let us again consider S and S′ to be in standard configuration (see Figure 1.1),and consider a photon emitted from the (coincident) origins of S and S′ at t =t′ = 0 and travelling in an arbitrary direction. Subsequently the space and timecoordinates of the photon in each frame must satisfy

c2t2−x2−y2− z2 = c2t′2−x′2−y′2− z′2 = 0�Substituting the relations (1.1) into this expression and solving for the constantsA and B, we obtain

ct′ = �ct−x��x′ = �x−ct��y′ = y�z′ = z�

(1.3)

where = v/c and = �1−2�−1/2. This Lorentz transformation, also knownas a boost in the x-direction, reduces to the Galilean transformation (1.2) when

� 1. Once again, symmetry demands that the unprimed coordinates are givenin terms of the primed coordinates by an analogous transformation in which v isreplaced by −v.

From the equations (1.3), we see that the time and space coordinates are ingeneral mixed by a Lorentz transformation (note, in particular, the symmetrybetween ct and x). Moreover, as we shall see shortly, if we consider two eventsA and B with coordinates �tA� xA� yA� zA� and �tB� xB� yB� zB� in S, it is straight-forward to show that the interval (squared)

�s2 = c2�t2−�x2−�y2−�z2 (1.4)

is invariant under any Lorentz transformation. As advocated by Minkowski, theseobservations lead us to consider space and time as united in a four-dimensionalcontinuum called spacetime, whose geometry is characterised by (1.4). We notethat the spacetime of special relativity is non-Euclidean, because of the minussigns in (1.4), and is often called the pseudo-Euclidean or Minkowski geometry.Nevertheless, for any fixed value of t the spatial part of the geometry remainsEuclidean.

2 The reasoning behind Einstein’s proposal is discussed in Appendix 1A.

1.4 Lorentz transformations as four-dimensional ‘rotations’ 5

We have arrived at the familiar viewpoint (to a physicist!) where the physicalworld is modelled as a four-dimensional spacetime continuum that possessesthe Minkowski geometry characterised by (1.4). Indeed, many ideas in specialrelativity are most simply explained by adopting a four-dimensional point of view.

1.4 Lorentz transformations as four-dimensional ‘rotations’

Adopting a particular (Cartesian) inertial frame S corresponds to labelling events inthe Minkowski spacetime with a given set of coordinates �t� x� y� z�. If we chooseinstead to describe the world with respect to a different Cartesian inertial frameS′ then this corresponds simply to relabelling events in the Minkowski spacetimewith a new set of coordinates �t′� x′� y′� z′�; the primed and unprimed coordinatesare related by the appropriate Lorentz transformation. Thus, describing physicsin terms of different inertial frames is equivalent to performing a coordinatetransformation on the Minkowski spacetime.

Consider, for example, the case where S′ is related to S via a spatial rotationthrough an angle � about the x-axis. In this case, we have

ct′ = ct�x′ = x′�y′ = y cos�− z sin ��z′ = y sin �+ z cos��

Clearly the inverse transform is obtained on replacing � by −�.The close similarity between the ‘boost’ (1.3) and an ordinary spatial rotation

can be highlighted by introducing the rapidity parameter

� = tanh−1�As varies from zero to unity, � ranges from 0 to�. We also note that = cosh�and = sinh�. If two inertial frames S and S′ are in standard configuration, wetherefore have

ct′ = ct cosh�−x sinh��x′ = −ct sinh�+x cosh��y′ = y�z′ = z�

(1.5)

This has essentially the same form as a spatial rotation, but with hyperbolicfunctions replacing trigonometric ones. Once again the inverse transformation isobtained on replacing � by −�.


x

y

S

z

z'

y'

x'

S'

a

v

Figure 1.2 Two inertial frames S and S′ in general configuration. The brokenline shown the trajectory of the origin of S′.

In general, S′ is moving with a constant velocity �v with respect to S in anarbitrary direction3 and the axes of S′ are rotated with respect to those of S.Moreover, at t = t′ = 0 the origins of S and S′ need not be coincident and maybe separated by a vector displacement �a, as measured in S (see Figure 1.2).4The corresponding transformation connecting the two inertial frames is mosteasily found by decomposing the transformation into a displacement, followedby a spatial rotation, followed by a boost, followed by a further spatial rotation.Physically, the displacement makes the origins of S and S′ coincident at t= t′ = 0,and the first rotation lines up the x-axis of S with the velocity �v of S′. Then a boostin this direction with speed v transforms S into a frame that is at rest with respect toS′. A final rotation lines up the coordinate frame with that of S′. The displacementand spatial rotations introduce no new physics, and the only special-relativisticconsideration concerns the boost. Thus, without loss of generality, we can restrictour attention to inertial frames S and S′ that are in standard configuration, forwhich the Lorentz transformation is given by (1.3) or (1.5).

1.5 The interval and the lightcone

If we consider two events A and B having coordinates �t′A�x′A� y′A� z′A� and�t′B� x′B� y′B� z′B� in S′, then, from (1.5), the interval between the events is given by

3 Throughout this book, the notation �v is used specifically to denote three-dimensional vectors, whereas vdenotes a general vector, which is most often a 4-vector.

4 If �a= �0 then the Lorentz transformation connecting the two inertial frames is called homogeneous, while if�a �= �0 it is called inhomogeneous. Inhomogeneous transformations are often referred to as Poincaré transfor-mations, in which case homogeneous transformations are referred to simply as Lorentz transformations.

1.5 The interval and the lightcone 7

�s2 = c2�t′2−�x′2−�y′2−�z′2

= �c�t� cosh�− ��x� sinh��2− −�c�t� sinh�+ ��x� cosh��2−�y2−�z2

= c2�t2−�x2−�y2−�z2�Thus the interval is invariant under the boost (1.5) and, from the above discussion,we may infer that �s2 is in fact invariant under any Poincaré transformation. Thissuggests that the interval is an underlying geometrical property of the spacetimeitself, i.e. an invariant ‘distance’ between events in spacetime. It also follows thatthe sign of �s2 is defined invariantly, as follows:

for �s2 > 0� the interval is timelike�for �s2 = 0� the interval is null or lightlike�for �s2 < 0� the interval is spacelike�

This embodies the standard lightcone structure shown in Figure 1.3. Events A andB are separated by a timelike interval, A and C by a lightlike (or null) interval and

ct

x

A

Future of A

Past of A

D

‘Elsewhere’ of A ‘Elsewhere’ of A

C

B

Figure 1.3 Spacetime diagram illustrating the lightcone of an event A (the y-and z- axes have been suppressed). Events A and B are separated by a timelikeinterval, A and C by a lightlike (or null) interval and A and D by a spacelikeinterval.


A and D by a spacelike interval. The geometrical distinction between timelike andspacelike intervals corresponds to a physical distinction: if the interval is timelikethen we can find an inertial frame in which the events occur at the same spatialcoordinates and if the interval is spacelike then we can find an inertial framein which the events occur at the same time coordinate. This becomes obviouswhen we consider the spacetime diagram of a Lorentz transformation; we shalldo this next.

1.6 Spacetime diagrams

Figure 1.3 is an example of a spacetime diagram. Such diagrams are extremelyuseful in illustrating directly many special-relativistic effects, in particular coor-dinate transformations on the Minkowski spacetime between different inertialframes. The spacetime diagram in Figure 1.4 shows the change of coordinates ofan event A corresponding to the standard-configuration Lorentz transformation(1.5). The x′-axis is simply the line t′ = 0 and the t′-axis is the line x′ = 0.From the Lorentz-boost transformation (1.3) we see that the angle between thex- and x′- axes is the same as that between the t- and t′- axes and has the value

t (A)

ct ct'

x

x'

t' (A)

x' (A)

Event A

x (A)

Figure 1.4 Spacetime diagram illustrating the coordinate transformationbetween two inertial frames S and S′ in standard configuration (the y- and z-axes have been suppressed). The worldlines of the origins of S and S′ are theaxes ct and ct′ respectively.

1.6 Spacetime diagrams 9

tan−1�v/c�. Moreover, we note that the t- and t′- axes are also the worldlines ofthe origins of S and S′ respectively.

It is important to realise that the coordinates of the event A in the frame S′ arenot obtained by extending perpendiculars from A to the x′- and t′- axes. Sincethe x′-axis is simply the line t′ = 0, it follows that lines of simultaneity in S′ areparallel to the x′-axis. Similarly, lines of constant x′ are parallel to the t′-axis. Thesame reasoning is equally valid for obtaining the coordinates of A in the frameS but, since the x- and t- axes are drawn as orthogonal in the diagram, this isequivalent simply to extending perpendiculars from A to the x- and t- axes in themore familiar manner.

The concept of simultaneity is simply illustrated using a spacetime diagram.For example, in Figure 1.5 we replot the events in Figure 1.3, together with the x′-and t′- axes corresponding to a Lorentz boost in standard configuration at somevelocity v. We see that the events A and D, which are separated by a spacelikeinterval, lie on a line of constant t′ and so are simultaneous in S′. Evidently, Aand D are not simultaneous in S; D occurs at a later time than A. In a similarway, it is straightforward to find a standard-configuration Lorentz boost such thatthe events A and B, which are separated by a timelike interval, lie on a line ofconstant x′ and hence occur at the same spatial location in S′.

line ofconstant t'

x'

ct'ct

x

A

D

C

B

Figure 1.5 The events illustrated in figure 1.3 and a Lorentz boost such that Aand D are simultaneous in S′.


1.7 Length contraction and time dilation

Two elementary (but profound) consequences of the Lorentz transformationsare length contraction and time dilation. Both these effects are easily derivedfrom (1.3).

Length contraction

Consider a rod of proper length �0 at rest in S′ (see Figure 1.6); we have

�0 = x′B−x′A�We want to apply the Lorentz transformation formulae and so find what lengthan observer in frame S assigns to the rod. Applying the second formula in (1.3),we obtain

x′A = �xA−vtA� �x′B = �xB−vtB� �

relating the coordinates of the ends of the rod in S′ to the coordinates in S. Theobserver in S measures the length of the rod at a fixed time t = tA = tB as

�= xB−xA =1

(x′B−x′A

)= �0�

Hence in S the rod appears contracted to the length

�= �0(1−v2/c2)1/2 �

If a rod is moving relative to S in a direction perpendicular to its length,however, it is straightforward to show that it suffers no contraction. It thus followsthat the volume V of a moving object, as measured by simultaneously noting thepositions of the boundary points in S, is related to its proper volume V0 by V =V0�1−v2/c2�1/2. This fact must be taken into account when considering densities.

x x'

y'y

S S'

z'z

v

x A' x B'

Figure 1.6 Two inertial frames S and S′ in standard configuration. A rod ofproper length �0 is at rest in S

′.

1.8 Invariant hyperbolae 11

‘Click 1’ ‘Click 2’

x'A x'Ax'

y'

S'

z'

x'

y'

S'

z'

x

y

S

z

vv

Figure 1.7 Two inertial frames S and S′ in standard configuration. A clock isat rest in S′.

Time dilation

Suppose we have a clock at rest in S′, in which two successive ‘clicks’ of theclock (events A and B) are separated by a time interval T0 (see Figure 1.7). Thetimes of the clicks as recorded in S are

tA = (t′A+vx′A/c2

)�

tB = (t′A+T0+vx′B/c2

)�

Since the clock is at rest in S′ we have x′A = x′B, and so on subtracting we obtain

T = tB− tA = T0 =T0

�1−v2/c2�1/2 �

Hence, the moving clock ticks more slowly by a factor of �1− v2/c2�1/2 (timedilation).

Note that an ideal clock is one that is unaffected by acceleration – externalforces act identically on all parts of the clock (an example is a muon).

1.8 Invariant hyperbolae

Length contraction and time dilation are easily illustrated using spacetimediagrams. However, while Figure 1.4 illustrates the positions of the x′- and t′- axescorresponding to a standard Lorentz boost, we have not yet calibrated the lengthscales along them. To perform this calibration, we make use of the fact that theinterval �s2 between two events is an invariant, and draw the invariant hyperbolae

c2t2−x2 = c2t′2−x′2 =±1on the spacetime diagram, as shown in Figure 1.8. Then, if we first take thepositive sign, setting ct = 0, we obtain x = ±1. It follows that OA is a unit


x'

ct'

O

line ofconstant x'

line ofconstant t'

C

D

ct

xA

B

Figure 1.8 The invariant hyperbolae c2t2−x2 = c2t′2−x′2 =±1.

distance along the x-axis. Now setting ct′ = 0 we find that x′ = ±1, so that OC isa unit distance along the x′-axis. Similarly, OB and OD are unit distances alongthe t- and t′- axes respectively. We also note that the tangents to the invarianthyperbolae at C and D are lines of constant x′ and t′ respectively.The length contraction and time dilation effects can now be read off directly

from the diagram. For example, the worldlines of the end-points of a unit rodOC in S′, namely x′ = 0 and x′ = 1, cut the x-axis in less than unit distance.Similarly, worldlines x = 0 and x = 1 in S cut the x′-axis inside OC, illustratingthe reciprocal nature of length contraction. Also, a clock at rest at the origin ofS′ will move along the t′-axis, reaching D with a reading of t′ = 1. However, theevent D has a t-coordinate that is greater than unity, thereby illustrating the timedilation effect.

1.9 The Minkowski spacetime line element

Let consider more closely the meaning of the interval between two events A andB in spacetime. Given that in a particular inertial frame S the coordinates of Aand B are �tA� xA� yA� zA� and �tB� xB� yB� zB�, we have so far taken the square ofthe interval between A and B to be

�s2 = c2�t2−�x2−�y2−�z2�

1.9 The Minkowski spacetime line element 13

ct

x

B

A

Figure 1.9 Two paths in spacetime connecting the events A and B.

where �t = tB− tA etc. This interval is invariant under Lorentz transformationand corresponds to the ‘distance’ in spacetime measured along the straight linein Figure 1.9 connecting A and B. This line may be interpreted as the worldlineof a particle moving at constant velocity relative to S between events A and B.However, the question naturally arises of what interval is measured between Aand B along some other path in spacetime, for example the ‘wiggly’ path shownin Figure 1.9.

To address this question, we must express the intrinsic geometry of theMinkowski spacetime in infinitesimal form. Clearly, if two infinitesimally sepa-rated events have coordinates �t� x� y� z� and �t+dt� x+dx� y+dy� z+dz� in Sthen the square of the infinitesimal interval between them is given by5

ds2 = c2dt2−dx2−dy2−dz2�which is known as the line element of Minkowski spacetime, or the special-relativistic line element. From our earlier considerations, it is clear that ds2 isinvariant under any Lorentz transformation. The invariant interval between A andB along an arbitrary path in spacetime is then given by

�s =∫ BA

ds�

5 To avoid mathematical ambiguity, one should properly denote the squares of infinitesimal coordinate intervalsby �dt�2 etc., but this notation is not in common use in relativity textbooks. We will thus adopt the moreusual form dt2, but it should be remembered that this is not the differential of t2.


where the integral is evaluated along the particular path under consideration.Clearly, to perform this integral we must have a set of equations describing thespacetime path.

1.10 Particle worldlines and proper time

Let us now turn to the description of the motion of a particle in spacetime terms.A particle describes a worldline in spacetime. In general, for two infinitesimallyseparated events in spacetime; by analogy with our earlier discussion we have:

for ds2 > 0� the interval is timelike�

for ds2 = 0� the interval is null or lightlike�for ds2 < 0� the interval is spacelike�

However, relativistic mechanics prohibits the acceleration of a massive particleto speeds greater than or equal to c, which implies that its worldline must liewithin the lightcone (Figure 1.3) at each event on it. In other words, the intervalbetween any two infinitesimally separated events on the particle’s worldline mustbe timelike (and future-pointing). For a massless particle such as a photon, anytwo events on its worldline are separated by a null interval. Figure 1.10 illustratesgeneral worldlines for a massive particle and for a photon.

ct

x

Figure 1.10 The worldlines of a photon (solid line) and a massive particle(broken line). The lightcones at seven events are shown.

1.10 Particle worldlines and proper time 15

A particle worldline may be described by giving x, y and z as functions of tin some inertial frame S. However, a more four-dimensional way of describing aworldline is to give the four coordinates �t� x� y� z� of the particle in S as functionsof a parameter � that varies monotonically along the worldline. Given the fourfunctions t��, x��, y�� and z��, each value of � determines a point alongthe curve. Any such parameter is possible, but a natural one to use for a massiveparticle is its proper time.We define the proper time interval d� between two infinitesimally separated

events on the particle’s worldline by

c2d�2 = ds2� (1.6)

Thus, if the coordinate differences in S between the two events are dt�dx�dy�dzthen we have

c2d�2 = c2dt2−dx2−dy2−dz2�

Hence the proper time interval between the events is given by

d� = �1−v2/c2�1/2dt = dt/v�

where v is the speed of the particle with respect to S over this infinitesimalinterval. If we integrate d� between two points A and B on the worldline, weobtain the total elapsed proper time interval:

�� =∫ BA

d� =∫ BA

[1− v

2�t�

c2

]1/2dt� (1.7)

We see that if the particle is at rest in S then the proper time � is just thecoordinate time t measured by clocks at rest in S. If at any instant in the historyof the particle we introduce an instantaneous rest frame S′ such that the particleis momentarily at rest in S′ then we see that the proper time � is simply thetime recorded by a clock that moves along with the particle. It is therefore aninvariantly defined quantity, a fact that is clear from (1.6).

Thus the worldline of a massive particle can be described by giving the fourcoordinates �t� x� y� z� as functions of � (see Figure 1.11). For example,

t = ��1−v2/c2�−1/2�x = v��1−v2/c2�−1/2�y = z= 0


τ

τ = –2

τ = –1

τ = 0

τ = 1

τ = 2

τ = 3

ct

x

Figure 1.11 A path in the �t� x�-plane can be specified by giving one coordinatein terms of the other, for example x = x�t�, or alternatively by giving bothcoordinates as functions of a parameter � along the curve: t = t�� x = x��.For massive particles the natural parameter to use is the proper time �.

is the worldline of a particle, moving at constant speed v along the x-axis of S,which passes through the origin of S at t = 0.

1.11 The Doppler effect

A useful illustration of particle worldlines and the concept of proper time isprovided by deriving the Doppler effect in a transparently four-dimensionalmanner. Let us consider an observer � at rest in some inertial frame S, and aradiation-emitting source � moving along the positive x-axis of S at a uniformspeed v. Suppose that the source emits the first wavecrest of a photon at anevent A, with coordinates �te� xe� in S, and the next wavecrest at an event Bwith coordinates �te+�te� xe+�xe�. Let us assume that these two wavecrestsreach the observer at the events C and D coordinates �to� xo� and �to+�to� xo�respectively. This situation is illustrated in Figure 1.12. From (1.7), the propertime interval experienced by � between the events A and B is

��AB =(1−v2/c2)1/2�te� (1.8)

and the proper time interval experienced by � between the events C and D is

��CD = �to� (1.9)

1.11 The Doppler effect 17

ε

D

CB

A

ct

x

Figure 1.12 Spacetime diagram of the Doppler effect.

Along each of the worldlines representing the photon wavecrests,

ds2 = c2 dt2−dx2−dy2−dz2 = 0�Thus, since we are assuming that dy = dz = 0, along the worldline connectingthe events A and C we have ∫ to

te

c dt =−∫ xoxe

dx� (1.10)

where the minus sign on the right-hand side arises because the photon is travellingin the negative x-direction. From (1.10), we obtain the (obvious) result c�to−te�=−�xo−xe�. Similarly, along the worldline connecting B and D we have∫ to+�to

te+�tec dt =−

∫ xoxe+�xe

dx�

Rewriting the integrals on each side, we obtain(∫ tote

+∫ to+�toto

−∫ te+�tete

)c dt =−

(∫ xoxe

−∫ xe+�xexe

)dx�

where the first integrals on each side of the equation cancel by virtue of (1.10).Thus we find that c�to− c�te = �xe, from which we obtain

�to =(1+ 1

c

�xe�te

)�te =

(1+ v

c

)�te� (1.11)


Hence, using (1.8), (1.9) and (1.11), we can derive the ratio of the proper timeintervals ��CD and ��AB experienced by � and � respectively:

��CD��AB

= �1+��te�1−2�1/2�te

= 1+

�1−�1/2�1+�1/2 =

�1+�1/2�1−�1/2 �

This ratio must be the reciprocal of the ratio of the photon’s frequency as measuredby � and � respectively, and thus we obtain the familiar Doppler-effect formula

��=(1−

1+

)1/2� (1.12)

1.12 Addition of velocities in special relativity

If a particle’s worldline is described by giving x, y and z as functions of t insome inertial frame S then the components of its velocity in S at any point are

ux =dx

dt� uy =

dy

dt� uz =

dz

dt�

The components of its velocity in some other inertial frame S′ are usually obtainedby taking differentials of the Lorentz transformation. For inertial frames S and S′related by a boost v in standard configuration, we have from (1.3)

dt′ = v�dt−vdx/c2�� dx′ = v�dx−vdt�� dy′ = dy� dz′ = dz�where we have made explicit the dependence of on v. We immediately obtain

u′x =dx′

dt′= ux−v

1−uxv/c2�

u′y =dy′

dt′= uy

v�1−uxv/c2��

u′z =dz′

dt′= uz

v�1−uxv/c2��

(1.13)

These replace the ‘common sense’ addition-of-velocities formulae of Newtonianmechanics. The inverse transformations are obtained by replacing v by −v.The special-relativistic addition of velocities along the same direction is

elegantly expressed using the rapidity parameter (Section 1.4). For example,consider three inertial frames S, S′ and S′′. Suppose that S′ is related to S by aboost of speed v in the x-direction and that S′′ is related to S′ by a boost of speedu′ in the x′-direction. Using (1.5), we quickly find that

ct′′ = ct cosh��v+�u′�−x sinh��v+�u′��

1.13 Acceleration in special relativity 19

x′′ = −ct sinh��v+�u′�+x cosh��v+�u′��y′′ = y�z′′ = z�

where tanh�v = v/c and tanh�u′ = u′/c. This shows that S′′ is connected to Sby a boost in the x-direction with speed u, where u/c = tanh��v+�u′�. Thus wesimply add the rapidities (in a similar way to adding the angles of two spatialrotations about the same axis). This gives

u= c tanh��v+�u′�= ctanh�v+ tanh�u′1+ tanh�v tanh�u′

= u′ +v

1+u′v/c2 �

which is the special-relativistic formula for the addition of velocities in the samedirection.

1.13 Acceleration in special relativity

The components of the acceleration of a particle in S are defined as

ax =duxdt

� ay =duy

dt� az =

duzdt

�

and the corresponding quantities in S′ are obtained from the differential forms ofthe expressions (1.13). For example,

du′x =dux

2v �1−uxv/c2�2�

Also, from the Lorentz transformation (1.3) we find that

dt′ = v�dt−vdx/c2�= v�1−uxv/c2�dt�So, for example, we have

a′x =du′xdt′

= 13v �1−uxv/c2�3

ax� (1.14)

Similarly, we obtain

a′y =du′ydt′

= 12v �1−uxv/c2�2

ay+uyv

c22v �1−uxv/c2�3ax

a′z =du′zdt′

= 12v �1−uxv/c2�2

az+uzv

c22v �1−uxv/c2�3ax


We see from these transformation formulae that acceleration is not invariantin special relativity, unlike in Newtonian mechanics, as discussed in Section 1.2.However, it is clear that acceleration is an absolute quantity, that is, all observersagree upon whether a body is accelerating. If the acceleration is zero in oneinertial frame, it is necessarily zero in any other frame.

Let us investigate the worldline of an accelerated particle. To make our illus-tration concrete, we consider a spaceship moving at a variable speed u�t� relativeto some inertial frame S and suppose that an observer B in the spaceship makesa continuous record of his accelerometer reading f�� as a function of his ownproper time �.

We begin by introducing an instantaneous rest frame (IRF) S′, which, at eachinstant, is an inertial frame moving at the same speed v as the spaceship, i.e. v= u.Thus, at any instant, the velocity of the spaceship in the IRF S′ is zero, i.e. u′ = 0.Moreover, from the above discussion of proper time, it should be clear that at anyinstant an interval of proper time is equal to an interval of coordinate time in theIRF, i.e. �� = �t′. An accelerometer measures the rate of change of velocity, sothat, during a small interval of proper time ��, B will record that his velocity haschanged by an amount f��. Therefore, at any instant, in the IRF S′ we have

du′

dt′= du

′

d�= f��

From (1.14), we thus obtain

du

dt=(1− u

2

c2

)3/2f��

However, since d� = �1−u2/c2�1/2 dt, we find thatdu

d�=(1− u

2

c2

)f��

which integrates easily to give

u��= c tanh��where c��= ∫ �0 f�� ′�d� ′ and we have taken u�� = 0� to be zero. Thus we havean expression for the velocity of the spaceship in S as a function of B’s proper time.

To parameterise the worldline of the spaceship in S, we note that

dt

d�=(1− u

2

c2

)−1/2= cosh��

dx

d�= u

(1− u

2

c2

)−1/2= c sinh�� (1.15)

Integration of these equations with respect to � gives the functions t�� and x��.

1.14 Event horizons in special relativity 21

1.14 Event horizons in special relativity

The presence of acceleration can produce surprising effects. Consider for simplic-ity the case of uniform acceleration. By this we mean we do not mean thatdu/dt = constant, since this is inappropiate in special relativity because it wouldimply that u→� as t→�, which is not permitted. Instead, uniform acceler-ation in special relativity means that the accelerometer reading f�� is constant.A spaceship whose engine is set at a constant emission rate would be uniformlyaccelerated in this sense.

Thus, if f = constant, we have � = f�/c. The equations (1.15) are then easilyintegrated to give

t = t0+c

fsinh

f�

c�

x = x0+c2

f

(cosh

f�

c−1

)�

where t0 and x0 are constants of integration. Setting t0 = x0 = 0 gives the pathshown in Figure 1.13. The worldline takes the form of a hyperbola.

Imagine that an observer B has the resources to maintain an acceleration findefinitely. Then there will be events that B will never be able to observe.The events in question lie on the future side of the asymptote to B’s hyperbola;this asymplote (which is a null line) is the event horizon of B. Objects whoseworldlines cross this horizon will disappear from B’s view and will seem to take

Horiz

on

B never sees Aafter this event

ct

x

BA

Figure 1.13 The worldline of a uniformly accelerated particle B starting fromrest from the origin of S. If an observer A remains at x = 0, then the worldlineof A is simply the t-axis. No message sent by A after t = c/f will ever reach B.


for ever to do so. Nevertheless, the objects themselves cross the horizon in a finiteproper time and still have an infinite lifetime ahead of them.

Appendix 1A: Einstein’s route to special relativity

Most books on special relativity begin with some sort of description of theMichelson–Morley experiment and then introduce the Lorentz transformation. Infact, Einstein claimed that he was not influenced by this experiment. This isdisputed by various historians of science and biographers of Einstein. One mightthink that these scholars are on strong ground, especially given that the experimentis referred to (albeit obliquely) in Einstein’s papers. However, it may be worthtaking Einstein’s claim at face value.

Remember that Einstein was a theorist – one of the greatest theorists who hasever lived – and he had a theorist’s way of looking at physics. A good theoristdevelops an intuition about how Nature works, which helps in the formulationof physical laws. For example, possible symmetries and conserved quantities areconsidered. We can get a strong clue about Einstein’s thinking from the title ofhis famous 1905 paper on special relativity. The first paragraph is reproducedbelow.

On the Electrodynamics of Moving Bodiesby A. Einstein

It is known that Maxwell’s electrodynamics – as usually understood at the present time –when applied to moving bodies, leads to asymmetries which do not appear to be inherentin the phenomena. Take, for example, the reciprocal electrodynamic action of a magnetand a conductor. The observable phenomenon here depends only on the relative motionof the conductor and the magnet, whereas the customary view draws a sharp distinctionbetween the two cases in which either the one or the other of these bodies is in motion.For if the magnet is in motion and the conductor at rest, there arises in the neighbourhoodof the magnet an electric field with a certain definite energy, producing a current at theplaces where parts of the conductor are situated. But if the magnet is stationary andthe conductor in motion, no electric field arises in the neighbourhood of the magnet. Inthe conductor, however, we find an electromotive force, to which in itself there is nocorresponding energy, but which gives rise – assuming equality of relative motion in thetwo cases discussed – to electric currents of the same path and intensity as those producedby the electric forces in the former case.

You see that Einstein’s paper is not called ‘Transformations between inertialframes’, or ‘A theory in which the speed of light is assumed to be a universalconstant’. Electrodynamics is at the heart of Einstein’s thinking; Einstein realizedthat Maxwell’s equations of electromagnetism required special relativity.

Appendix 1A: Einstein’s route to special relativity 23

Maxwell’s equations are

�� · �D = �� · �B = 0�

��× �E =−��B�t

� ��× �H = �j+ � �D�t

�

where �D= �0 �E+ �P and �B= �0� �H+ �M�, �P and �M being respectively the polari-sation and the magnetisation of the medium in which the fields are present. In freespace we can set �j= �0 and �= 0, and we then get the more obviously symmetricalequations

�� · �E = 0� �� · �B = 0�

��× �E =−��B�t

� ��× �B = �0�0� �E�t

�

Taking the curl of the equation for ��× �E, applying the relation��× ��× �E�= �� · �E�−�2 �E

and performing a similar operation for �B in the equation for ��× �B, we derive theequations for electromagnetic waves:

�2 �E = �0�0�2 �E�t2

� �2 �B = �0�0�2 �B�t2

�

These both have the form of a wave equation with a propagation speed c =1/√�0�0. Now, the constants �0 and �0 are properties of the ‘vacuum’:

�0� the permeability of a vacuum, equals 4�×10−7 Hm−1��0� the permittivity of a vacuum, equals 8�85×10−12 Fm−1�

This relation between the constants �0 and �0 and the speed of light was one ofthe most startling consequences of Maxwell’s theory. But what do we mean by a‘vacuum’? Does it define an absolute frame of rest? If we deny the existence of anabsolute frame of rest then how do we formulate a theory of electromagnetism?How do Maxwell’s equations appear in frames moving with respect to each other?Do we need to change the value of c? If we do, what will happen to the valuesof �0 and �0?

Einstein solves all of these problems at a stroke by saying that Maxwell’sequations take the same mathematical form in all inertial frames. The speed of lightc is thus the same in all inertial frames. The theory of special relativity (includingamazing conclusions such as E = mc2) follows from a generalisation of thissimple and theoretically compelling assumption. Maxwell’s equations thereforerequire special relativity. You see that for a master theorist like Einstein, the


Michelson–Morley experiment might well have been a side issue. Einstein could‘see’ special relativity lurking in Maxwell’s equations.

Exercises

1.1 For two inertial frames S and S′ in standard configuration, show that the coordinatesof any given event in each frame are related by the Lorentz tranformations (1.3).

1.2 Two events A and B have coordinates �tA� xA� yA� zA� and �tB� xB� yB� zB� respectively.Show that both the time difference �t = tB− tA and the quantity

�r2 = �x2+�y2+�z2

are separately invariant under any Galilean transformation, whereas the quantity

�s2 = c2�t2−�x2−�y2−�z2

is invariant under any Lorentz transformation.1.3 In a given inertial frame two particles are shot out simultaneously from a given

point, with equal speeds v in orthogonal directions. What is the speed of each particlerelative to the other?

1.4 An inertial frame S′ is related to S by a boost of speed v in the x-direction, and S′′

is related to S′ by a boost of speed u′ in the x′-direction. Show that S′′ is related toS by a boost in the x-direction with speed u, where

u= c tanh��v+�u′��tanh�v = v/c and tanh�u′ = u′/c.

1.5 An inertial frame S′ is related to S by a boost �v whose components in S are �vx� vy� vz�.Show that the coordinates �ct′� x′� y′� z′� and �ct� x� y� z� of an event are related by⎛⎜⎜⎜⎝

ct′

x′

y′

z′

⎞⎟⎟⎟⎠=⎛⎜⎜⎜⎝

−x −y −z−x 1+�2x �xy �xz−y �yx 1+�2y �yz−y �zx �zy 1+�2z

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝ct

x

y

z

⎞⎟⎟⎟⎠ �

where � = �v/c� = �1−��2�−1/2 and � = �− 1�/��2. Hint: The transformationmust take the same form if both S and S′ undergo the same spatial rotation.

1.6 An inertial frame S′ is related to S by a boost of speed u in the positive x-direction.Similarly, S′′ is related to S′ by a boost of speed v in the y′-direction. Find thetransformation relating the coordinates �ct� x� y� z� and �ct′′� x′′� y′′� z′′� and hencedescribe how S and S′′ are physically related.

1.7 The frames S and S′ are in standard configuration. A straight rod rotates at a uniformangular velocity �′ about its centre, which is fixed at the origin of S′. If the rod liesalong the x′-axis at t′ = 0, obtain an equation for the shape of the rod in S at t = 0.

Exercises 25

1.8 Two events A and B have coordinates �tA� xA� yA� zA� and �tB� xB� yB� zB� respec-tively in some inertial frame S and are separated by a spacelike interval. Obtain anexpression for the boost �v required to transform to a new inertial frame S′ in whichthe events A and B occur simultaneously.

1.9 Derive the Doppler effect (1.12) directly, using the Lorentz transformationformulae (1.3).

1.10 Two observers are moving along trajectories parallel to the y-axis in some inertialframe. Observer A emits a photon with frequency �A that travels in the positivex-direction and is received by observer B with frequency �B. Show that the Dopplershift �B/�A in the photon frequency is the same whether A and B travel in the samedirection or opposite directions.

1.11 Astronauts in a spaceship travelling in a straight line past the Earth at speed v= c/2wish to tune into Radio 4 on 198 kHz. To what frequency should they tune at theinstant when the ship is closest to Earth?

1.12 Draw a spacetime diagram illustrating the coordinate transformation correspondingto two inertial frames S and S′ in standard configuration (i.e. where S′ moves at aspeed v along the positive x-direction and the two frames coincide at t = t′ = 0).Show that the angle between the x- and x′- axes is the same as that between the t-and t′- axes and has the value tan−1�v/c�.

1.13 Consider an event P separated by a timelike interval from the origin O of yourdiagram in Exercise 1.12. Show that the tangent to the invariant hyperbola passingthrough P is a line of simultaneity in the inertial frame whose time axis joins Pto the origin. Hence, from your spacetime diagram, derive the formulae for lengthcontraction and time dilation.

1.14 Alex and Bob are twins working on a space station located at a fixed position indeep space. Alex undertakes an extended return spaceflight to a distant star, whileBob stays on the station. Show that, on his return to the station, the proper timeinterval experienced by Alex must be less than that experienced by Bob, hence Bobis now the elder. How does Alex explain this age difference?

1.15 A spaceship travels at a variable speed u�t� in some inertial frame S. An observeron the spaceship measures its acceleration to be f��, where � is the proper time.If at � = 0 the spaceship has a speed u0 in S show that

u��−u01−u��u0/c2

= c tanh��

where c��= ∫ �0 f�� ′�d� ′. Show that the velocity of the spaceship can never reach c.1.16 If the spaceship in Exercise 1.15 left base at time t = � = 0 and travelled forever

in a straight line with constant acceleration f , show that no signal sent by baselater than time t = c/f can ever reach the spaceship. By sketching an appropriatespacetime diagram show that light signals sent from the base appear increasinglyredshifted to an observer on the spaceship. If the acceleration of the spaceship is g(for the comfort of its occupants), how long by the spaceship clock does it take toreach a star 10 light years from the base?

2

Manifolds and coordinates

Our discussion of special relativity has led us to model the physical world as afour-dimensional continuum, called spacetime, with a Minkowski geometry. Thisis an example of a manifold. As we shall see, the more complicated spacetimegeometries of general relativity are also examples of manifolds. It is thereforeworthwhile discussing manifolds in general. In the following we consider generalproperties of manifolds commonly encountered in physics, and we concentrate inparticular on Riemannian manifolds, which will be central to our discussion ofgeneral relativity.

2.1 The concept of a manifold

In general, a manifold is any set that can be continuously parameterised. Thenumber of independent parameters required to specify any point in the set uniquelyis the dimension of the manifold, and the parameters themselves are the coor-dinates of the manifold. An abstract example is the set of all rigid rotations ofCartesian coordinate systems in three-dimensional Euclidean space, which can beparameterised by the Euler angles. So the set of rotations is a three-dimensionalmanifold: each point is a particular rotation, and the coordinates of the pointare the three Euler angles. Similarly, the phase space of a particle in classicalmechanics can be parameterised by three position coordinates �q1� q2� q3� andthree momentum coordinates �p1� p2� p3�, and thus the set of points in this phasespace forms a six-dimensional manifold. In fact, one can regard ‘manifold’ as justa fancy word for ‘space’ in the general mathematical sense.

In its most primitive form a general manifold is simply an amorphous collectionof points. Most manifolds used in physics, however, are ‘differential manifolds’,which are continuous and differentiable in the following way. A manifold iscontinuous if, in the neighbourhood of every point P, there are other points whosecoordinates differ infinitesimally from those of P. A manifold is differentiable ifit is possible to define a scalar field at each point of the manifold that can bedifferentiated everywhere. Both our examples above are differential manifolds.

26

2.3 Curves and surfaces 27

The association of points with the values of their parameters can be thought ofas a mapping of the points of a manifold into points of the Euclidean space of thesame dimension. This means that ‘locally’ a manifold looks like the correspondingEuclidean space: it is ‘smooth’ and has a certain number of dimensions.

2.2 Coordinates

An N -dimensional manifold � of points is one for which N independent realcoordinates �x1� x2� � � � � xN � are required to specify any point completely.1 TheseN coordinates are entirely general and are denoted collectively by xa, where it isunderstood that a= 1�2� � � � �N .As a technical point, we should mention that in general it may not be possible

to cover the whole manifold with only one non-degenerate coordinate system,namely, one which ascribes a unique set of N coordinate values to each point,so that the correspondence between points and sets of coordinate values (labels)is one-to-one. Let us consider, for example, the points that constitute a plane.These points clearly form a two-dimensional manifold (called R2). An exampleof a degenerate coordinate system on this manifold is the polar coordinates �r��in the plane, which have a degeneracy at the origin because � is indeterminatethere. For this manifold, we could avoid the degeneracy at the origin by using,for example, Cartesian coordinates. For a general manifold, however, we mighthave no choice in the matter and might have to work with coordinate systems thatcover only a portion of the manifold, called coordinate patches. For example, theset of points making up the surface of a sphere forms a two-dimensional manifold(called S2). This manifold is usually ‘parameterised’ by the coordinates � and�, but � is degenerate at the poles. In this case, however, it can be shown thatthere is no coordinate system that covers the whole of S2 without degeneracy; thesmallest number of patches needed is two. In general, a set of coordinate patchesthat covers the whole manifold is called an atlas.

Thus, in general, we do not require the whole of a manifold � to be coveredby a single coordinate system. Instead, we may have a collection of coordinatesystems, each covering some part of � , and all these are on an equal footing.We do not regard any one coordinate system as in some way preferred.

2.3 Curves and surfaces

Given a manifold, we shall be concerned with points in it and with subsets ofpoints that define curves and surfaces. We shall frequently define these curves

1 The reason why the coordinates are written with superscripts rather than subscripts will become clear later.

28 Manifolds and coordinates

and surfaces parametrically. Thus, since a curve has one degree of freedom,it depends on one parameter and so we define a curve in the manifold by theparametric equations

xa = xa�u� �a= 1�2� � � � �N��

where u is some parameter and x1�u�� x2�u�� xN �u� denote N functions of u.Similarly, since a submanifold or surface of M dimensions �M < N� has M

degrees of freedom, it depends on M parameters and is given by the N parametricequations

xa = xa�u1� u2� � � � � uM� �a= 1�2� � � � �N�� (2.1)

If, in particular, M = N −1 then the submanifold is called a hypersurface. In thiscase, the N −1 parameters can be eliminated from these N equations to give oneequation relating the coordinates, i.e.

f�x1� x2� � � � � xN �= 0�From a different but equivalent point of view, a point in a manifold is charac-

terised by N coordinates. If the point is restricted to lie in a particular hypersurface,i.e. an �N − 1�-dimensional subspace, then the point’s coordinates must satisfyone constraint equation, namely

f�x1� x2� � � � � xN �= 0�Similarly, points in an M-dimensional subspace �M < N� must satisfy N −Mconstraints

f1�x1� x2� � � � � xN � = 0�

f2�x1� x2� � � � � xN � = 0�

��

fN−M�x1� x2� � � � � xM� = 0�which is an alternative to the parametric representation (2.1).

2.4 Coordinate transformations

To locate a point in a manifold we use a system of N coordinates, but the choice ofthese coordinates is arbitrary. The important idea is not the ‘labels’ but the pointsthemselves and the geometrical and topological relationships between them.

2.4 Coordinate transformations 29

We may relabel the points of a manifold by performing a coordinate transfor-mation xa→ x′a expressed by the N equations

x′a = x′a�x1� x2� � � � � xN � �a= 1�2� � � � �N�� (2.2)

giving each new coordinate as a function of the old coordinates. Hence weview a coordinate transformation passively as assigning the new primed coor-dinates �x′1� x′2� � � � � x′N � to a point of the manifold whose old coordinates are�x1� x2� � � � � xN �.

We will assume that the functions involved in (2.2) are single-valued, contin-uous and differentiable over the valid ranges of their arguments. Thus by differ-entiating each equation in (2.2) with respect to each of the old coordinates xb weobtain the N ×N partial derivatives �x′a/�xb. These may be assembled into theN ×N transformation matrix2

[�x′a

�xb

]=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

�x′1

�x1�x′1

�x2· · · �x

′1

�xN

�x′2

�x1�x′2

�x2· · · �x

′2

�xN

��

��

�x′N

�x1�x′N

�x2· · · �x

′N

�xN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠�

so that rows are labelled by the index in the numerator of the partial derivativeand columns by the index in the denominator. The elements of the transforma-tion matrix are functions of the coordinates, and so the numerical values of thematrix elements are in general different when evaluated at different points in themanifold. The determinant of the transformation matrix is called the Jacobian ofthe transformation and is denoted by

J = det[�x′a

�xb

]�

Clearly, the numerical value of J also varies from point to point in the manifold.If J �= 0 for some range of the coordinates xb then it follows that in this region

we can (in principle) solve the equations (2.2) for the old coordinates xb andobtain the inverse transformation equations

xa = xa�x′1� x′2� � � � � x′N � �a= 1�2� � � � �N��

2 In general the notation � denotes the matrix containing the elements within the square brackets.

30 Manifolds and coordinates

In a similar manner to the above, we define the inverse transformation matrix

�xa/�x′b� and the Jacobian of the inverse transformation J ′ = det�xa/�x′b�.

Using the chain rule, it is easy to show that the inverse transformation matrixis the inverse of the transformation matrix, since

N∑b=1

�x′a

�xb�xb

�x′c= �x

′a

�x′c= �ac =

{1 if a= c�0 if a �= c�

where we have defined the Kronecker delta �ac and used the fact that

�x′a

�x′c= �x

a

�xc= 0 if a �= c�

because the coordinates in either the unprimed or the primed set are independent.Since the two transformation matrices are inverses of one another, it follows thatJ ′ = 1/J .

If we consider neighbouring points P and Q in the manifold, with coordinatesxa and xa+ dxa respectively, then in the new, primed, coordinate system theinfinitesimal coordinate separation between P and Q is given by

dx′a = �x′a

�x1dx1+ �x

′a

�x2dx2+· · ·+ �x

′a

�xNdxN �

where it is understood that the partial derivatives on the right-hand side areevaluated at the point P. We can write this more economically as

dx′a =N∑b=1

�x′a

�xbdxb� (2.3)

2.5 Summation convention

Our notation can be made more economical still by adopting Einstein’s summationconvention: whenever an index occurs twice in an expression, once as a subscriptand once as a superscript, this is understood to imply a summation over the indexfrom 1 to N , the dimension of the manifold.

Thus we can write (2.3) simply as

dx′a = �x′a

�xbdxb�

where, once again, it is understood that all the partial derivatives are evaluated atP. The index a appearing on each side of this equation is said to be a free indexand may take on separately any value from 1 to N . We consider a superscript that

2.6 Geometry of manifolds 31

appears in the denominator of a partial derivative as a subscript (and vice versa).Thus the index b on the right-hand side in effect appears once as a subscriptand once as a superscript, and hence there is an implied summation from 1 toN . An index that is summed over in this way is called a dummy index, becauseit can be replaced by any other index not already in use. For example, we maywrite

�x′a

�xbdxb = �x

′a

�xcdxc�

since c was not already in use in the expression.Note that the proper use of the summation convention requires that, in any

term, an index should not occur more than twice and that any repeated index mustoccur once as a subscript and once as a superscript.

2.6 Geometry of manifolds

So far, we have considered manifolds only in a very primitive form. We haveassumed that the manifold is continuous and differentiable, but aside from theseproperties it remains an amorphous collection of points. We have not yet definedits geometry.

Consider two infinitesimally separated points P and Q in the manifold, withcoordinates xa and xa+dxa respectively �a = 1�2� � � � �N�. The local geometryof the manifold at the point P is determined by defining the

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